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By Alfred Barnard Basset

This quantity is made from electronic photographs from the Cornell collage Library ancient arithmetic Monographs assortment.

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T h u s one gets the First Bianchi Identity Rijki + Rkju + Rijik = 0. 4), we have ^2 Pijkaul A ujk A u>na = 0 modw'Aw'Aw'. Hence the Minkowskian curvature satisfies t h a t * i ka = *k ia- \p-' ) Setting u>ij = ^2SjkU>ik. 8) Riemann Invariants 55 where {Hija} is the Cartan tensor of (M, F). 8), we have dun + dun = - 2 ^2(dHija) A w£ - 2 5 3 Hijadw%. 9) a Put ilij = ^25jk£lik. • w„" = 0). 10) where (-0 = 53Wifc A Wfcj + 5 3 ^ A Wki +2 y^AdHv

8, w1 A • • • A u>n A w\n A • • • A un-\n be considered as a volume form on SM. Setting I I : = SijLO1 UJJ + SapU)na ca (g> LJn13. Then S M is a (2n-l)-dimensional Riemannian manifold with Sasaki-type Riemannian metric G[Bao and Shen, 1994]. 24). 8, we put d(FFyiyi) = Kijau>an + GijkW , where K^a and Gijk are symmetric with respect to i and j . 30), we choose respectively ApCT = - ^ u/u^idjn VP*a = -^^(ufj-uJGijv + FyJX, + FyiXJ), - ujujGitf + uJu^Giji). 57) where n Chern Connection 33 and (_d_ _d_ ~ \dx^dx^dx^ d Aijk A Proof.

For functions a1 — ai{x), we have where n I dV := y/det{gij) ^ {-\)k-1ykdy1 A • • • A dyk A • • • A dyn. 5([Bao and Chern, 1996]) manifold. Then Vol(x)=constant. Let (M,F) be a weak Landsberg An Introduction 44 Example to Finsler Geometry Define F : TR2 -> [0, oo) by := ^p* + q2 + \(x)(p4 + q*) F(x,y) where y = (p, q) € TXR2 and A is a smoothly non-negative function. -> - I Wv)-\ \ (cf. 2). Hence , Ap 2 (p 4 +3q 4 ) -2ApV (P 4 +9 4 )* _2APV (p4+q4)I y/det{gij) = J:1 + A (p 4 +? 4 )5 1 | \ _ 1 A,2(3p4+g4) I (p4+q4)t / ^JV)1 + 4 4 (p +

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